Puzzle for January 30, 2021 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
In eq.2, replace D with A + C (from eq.4): C + A + C = A + B which becomes A + 2×C = A + B Subtract A from each side of the equation above: A + 2×C – A = A + B – A which makes 2×C = B
Hint #2
eq.5 may be written as: D – E = (A + F) ÷ 2 Multiply both sides of the above equation by 2: (D – E) × 2 = (A + F) ÷ 2 × 2 which becomes eq.5a) 2×D – 2×E = A + F
Hint #3
eq.6 may be written as: B = (A + C + D + F) ÷ 4 Multiply both sides of the above equation by 4: B × 4 = (A + C + D + F) ÷ 4 × 4 which becomes 4×B = A + C + D + F which may be written as eq.6a) 4×B = A + F + C + D
Hint #4
In eq.6a, replace B with (2×C), and A + F with 2×D – 2×E (from eq.5a): 4×(2×C) = 2×D – 2×E + C + D which becomes 8×C = 3×D – 2×E + C Subtract C from both sides of the above equation: 8×C – C = 3×D – 2×E + C – C which becomes eq.6b) 7×C = 3×D – 2×E
Hint #5
Add D and E to both sides of eq.3: E – D + D + E = C – E + D + E which becomes eq.3a) 2×E = C + D In eq.6b, substitute (C + D) for 2×E: 7×C = 3×D – (C + D) which becomes 7×C = 3×D – C – D which becomes 7×C = 2×D – C Add C to both sides of the equation above: 7×C + C = 2×D – C + C which becomes 8×C = 2×D Divide both sides by 2: 8×C ÷ 2 = 2×D ÷ 2 which makes 4×C = D
Hint #6
Substitute 4×C for D in eq.3a: 2×E = C + 4×C which makes 2×E = 5×C Divide both sides of the equation above by 2: 2×E ÷ 2 = 5×C ÷ 2 which makes E = 2½×C
Hint #7
eq.1 may be written as: A + F + B + C + D + E = 25 Substitute 2×D – 2×E for A + F (from eq.5a) in the equation above: 2×D – 2×E + B + C + D + E = 25 which becomes eq.1a) 3×D – E + B + C = 25
Hint #8
Substitute (4×C) for D, 2½×C for E, and 2×C for B in eq.1a: 3×(4×C) – 2½×C + 2×C + C = 25 which becomes 12×C + ½×C = 25 which becomes 12½×C = 25 Divide both sides of the above equation by 12½: 12½×C ÷ 12½ = 25 ÷ 12½ which means C = 2 making B = 2×C = 2 × 2 = 4 D = 4×C = 4 × 2 = 8 E = 2½×C = 2½ × 2 = 5
Hint #9
Substitute 2 for C, 8 for D, and 4 for B in eq.2: 2 + 8 = A + 4 which becomes 10 = A + 4 Subtract 4 from both sides of the above equation: 10 – 4 = A + 4 – 4 which makes 6 = A
Solution
Substitute 8 for D, 5 for E, and 6 for A in eq.5a: 2×8 – 2×5 = 6 + F which becomes 16 – 10 = 6 + F which becomes 6 = 6 + F Subtract 6 from each side of the equation above: 6 – 6 = 6 + F – 6 which makes 0 = F and makes ABCDEF = 642850